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J Control Theory Appl 2008 6 (1) 59–68

DOI 10.1007/s11768-008-7193-9

A novel induction motor control scheme

using IDA-PBC

Humberto GONZ

ALEZ

, Manuel A. DUARTE-MERMOUD

, Ian PELISSIER

Juan Carlos TRAVIESO-TORRES

, Romeo ORTEGA

(1.Department of Electrical Engineering, University of Chile, Casilla 412-3, Santiago, Chile;

2.Laboratoire des Signaux et Syst

emes, Plateau de Moulon 91192, Gif-sur-Yvette, France)

Abstract: A new control scheme for induction motors is proposed in the present paper, applying the interconnection

and damping assignment-passivity based control (IDA-PBC) method. The scheme is based exclusively on passivity based

control, without restricting the input frequency as it is done in ﬁeld oriented control (FOC). A port-controlled Hamiltonian

(PCH) model of the induction motor is deduced to make the interconnection and damping of energy explicit on the scheme.

The proposed controller is validated under computational simulations and experimental tests using an inverter prototype.

Keywords: Induction motor; Passivity based control; Energy shaping control; Interconnection and damping assign-

ment; IDA-PBC; Port-controlled Hamiltonian; PCH

1 Introduction

Passivity-based control (PBC) has become an important

tool in nonlinear control research, mainly because of its

straightforward application to physical systems (mechani-

cal, electrical and electromechanical). In recent years, inter-

connection and damping assignment-passivity based con-

trol (IDA-PBC) has appeared as a ﬂexible and versatile

method to design controllers for nonlinear systems, intro-

ducing tools to assign the interconnection and damping of

internal energy.

Also, induction motors are an interesting area of research

for nonlinear control [1, 2]. Their wider use replacing DC

motors has been an incentive to develop more and better

techniques to control their dynamic behavior to obtain a

more efﬁcient use of the energy without losing performance.

The present paper analyzes the use of IDA-PBC as a

method to control induction motors. The result is a new

scheme to design controllers for induction motors focus-

ing on energy characteristics, like equilibrium and shape, to

achieve the desired objective. This result does not use ﬁeld

oriented control (FOC) or any other scheme to ﬁx the in-

put voltage frequency; on the contrary, the input frequency

is used as a fundamental part of the solution. As a result

of the new scheme, a speed regulator is obtained and simu-

lated with an induction motor model in the Matlab-Simulink

environment. To apply the IDA-PBC method to induction

motors, a port-controlled Hamiltonian (PCH) model is de-

ducted for the complete electro-mechanical system. Also,

using an inverter prototype designed in [3], a set of experi-

ments are performed showing the behavior of the proposed

scheme.

This IDA-PBC controller is based on a different paradigm

for the induction motor control than the one used in FOC.

Instead of decoupling the inputs of the induction motor to

make it similar to a DC motor, the IDA-PBC controller takes

advantage of the internal properties of the system to reach

the desired objective without intermediate steps.

2 IDA-PBC control

2.1 Passive systems

The deﬁnition of a passive system is taken from [4]. Other

ways to state this deﬁnition, using the concept of dissipative

systems, can be found in [5].

Deﬁnition 1 [4] Let Σ

be a dynamical system,



˙x = f(x)+g(x)u,

y = h(x)

(1)

with u, y ∈ R

, x ∈ X ⊆ R

and an equilibrium point

∗

∈ X such that f(x

∗

)=0and h(x

∗

)=0. This sys-

tem will be called passive if there is a continuous function

H : X → R

with H(x

∗

)=0, called the storage function,

such that ∀t  t

∈ R and ∀u(·)

H(x(t)) − H(x(t

)) 



(s)y(s)ds. (2)

The function H(·) is related with the stored energy of a

system and, in a consistent way, the product u

(t)y(t) has

instant power units. The inequality (2) shows that the inter-

nal stored energy of a passive system is always less than or

equal to the energy supplied to it, or in other words, a pas-

Received 22 October 2007.

This work was supported by CONICYT Chile through grant FONDECYT (1061170).

60 H. GONZALEZ et al. / J Control Theory Appl 2008 6 (1) 59–68

sive system is unable to generate energy. The relation (2) is

called Dissipation Inequality.

2.2 PCH models

An interesting point of view in modeling dynamical sys-

tems has been raised following the known methodologies

of Lagrange and Hamilton, because they generate equations

with structures that allow to establish physical relationships

based on their variables and parameters. Particularly, we

will analyze the systems modeled with the Hamilton equa-

tions. A deeper study on this issues is found on [5∼8].

Deﬁnition 2 [5] A dynamical system Σ

PCH

has a PCH

model if its mathematic representation has the form,

PCH



˙x =[J(x) −R(x)] ∇H + g(x)u,

y = g

(x)∇H,

(3)

where

· H(x):R

→ R is a C

function that represents the in-

ternal energy in the system.

· x ∈ R

are the state variables.

·J(x)=−J

(x) is the interconnection matrix.

·R(x)=R(x)  0 is the damping matrix.

· g(x) is the input matrix.

The close relationship between the physical phenomenon

and its PCH model is the main contribution of this represen-

tation, because it allows a direct read of the internal inter-

connection and damping of energy on the matrices J and

R. This reason makes the PCH models particulary suitable

to analyze electrical, mechanical or electromechanical sys-

tems, because the physical knowledge is used to understand

their behaviour.

Moreover, it is proved in [5] that if H(·) is bounded from

below, then the system Σ

PCH

is passive, with H(·) as the

storage function.

2.3 IDA-PBC control

The interconnection and damping assignment-passivity-

based control (IDA-PBC), when used on PCH models, can

assign the interconnection, damping and internal energy of

a closed loop by changing the matrices J, Rand the storage

function H. The formulation for general models of systems

is found in [9, 10].

Theorem 1 Let Σ

PCH

be a system described by the

equation (3). Let us assume that there are matrices g

⊥

(x),

= −J

, R

= R

 0 and a function H

: R

→ R

such that

⊥

[J−R] ∇H = g

⊥

−R

] ∇H

, (4)

where g

⊥

(x) is a full range left-annihilator of g(x), i.e.,

⊥

g =0, and H

such that

∗

=argmin

x∈R

(x). (5)

Then, applying the control u = β(x), where

β =





−1

{[J

−R

] ∇H

−[J−R] ∇H}, (6)

the closed-loop dynamics will be

˙x =[J

−R

] ∇H

(7)

with x

∗

as a locally stable equilibrium point in the Lyapunov

sense. The equilibrium point will be asymptotically stable in

the Lyapunov sense if the largest invariant set contained in



x ∈ R



∇H

(x)R

(x)∇H

(x)



(8)

equals {x

∗

A complete demonstration of the previous theorem can

be found in [10].

3 Induction motor: PCH model and IDA-

PBC control

The deduction of the PCH model for an induction motor

will be divided in two parts: the electrical subsystem model

and the mechanical subsystem model.

3.1 Electrical subsystem

The general equations for AC machines are used to study

this subsystem, as they are commonly used in [11, 12] to

analyze triphase motors.

Under symmetry hypothesis of the coils in the motor, and

using the Voltage Kirchhoff law over stator and rotor cir-

cuits, the equations are

= R

dψ

, (9a)

0=R

dψ

, (9b)

with u

being the voltage applied to the stator, i

and i

the

stator and rotor currents respectively, ψ

and ψ

the mag-

netic ﬂuxes linked by the stator and rotor coils, and R

and

the stator and rotor internal resistance, respectively. In

addition, the magnetic coupling between stator and rotor is

modeled as

= L

+ L

,ψ

= L

+ L

, (10)

where L

and L

are the stator and rotor self-inductances,

and L

is the mutual-inductance between both circuits.

Let a new reference frame rotated ρ

radians with respect

to the stator frame, as shown in Fig.1.

Fig. 1 Phasor diagram of the reference frame rotated ρ

radians.

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